- #1
gabu
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Homework Statement
Using the Schrödinger equation find the parameter [itex]\alpha[/itex] of the Harmonic Oscillator solution [itex]\Psi(x)=A x e^{-\alpha x^2}[/itex]
Homework Equations
[itex]-\frac{\hbar^2}{2m}\,\frac{\partial^2 \Psi(x)}{\partial x^2} + \frac{m \omega^2 x^2}{2}\Psi(x)=E\Psi(x)[/itex]
[itex]E=\hbar\omega(n+\frac{1}{2})[/itex]
The Attempt at a Solution
Using the Schrödinger equation we have arrive at
[itex] -2\alpha (2x^2\alpha-3)+\frac{m^2\omega^2x^2}{\hbar^2} = \frac{2m}{\hbar^2}E[/itex]
If I make x=0 I obtain
[itex] \alpha = \frac{m\omega}{2\hbar} [/itex]
using the information that the energy level of the oscillator is the same as the highest power in the solution, meaning [itex] E=3\hbar\omega/2 [/itex].
Now, my problem with this solution is the need to make x=0 to arrive at it. I know that the equation holds for every x, so it is justifiable to consider the origin. The thing is, however, that shouldn't I be able to solve the equation without this assumption? Shouldn't it be independent of x?
Thank you very much.